The dimer model on the generalized checkerboard lattice — a square lattice whose four-site unit cell carries alternating bond weights x1,x2x_1,x_2 (horizontal) and y1,y2y_1,y_2 (vertical) — has three inequivalent weight assignments, denoted checkerboard A, B, and C. Izmailian, Hu, and Kenna (2015) derived the exact closed-form partition functions of all three models on the 2M×2N2M\times2N torus with periodic boundary conditions and showed that checkerboard B and C interpolate between a c=−2c=-2 rectangular-dimer regime and c=1c=1 Kasteleyn (anisotropic honeycomb) regimes with KDP-type singularities at the phase boundaries y1=x1+x2y_1=x_1+x_2 and permutations, explicitly leaving the finite-size analysis of these formulas as follow-up work. The finite-size corrections and scaling functions of the checkerboard-A lattice were subsequently derived exactly (2016), and the checkerboard-B lattice received a detailed numerical study of its specific heat and partition-function zeros (2019). The analytic finite-size theory of the B and C lattices remains open: exact asymptotic expansions of the free energy with leading and subleading correction amplitudes, crossover scaling functions connecting the c=−2c=-2 and c=1c=1 regimes, and explicit laws for the finite-size rounding and shift of the KDP-type specific-heat singularities and Fisher zeros.

Background

A close-packed dimer model places a bond (a 'dimer') so that every site of a lattice is covered exactly once, with each bond carrying a Boltzmann weight. Kasteleyn (1963) showed that the partition function of planar dimer models can be written as a Pfaffian — a signed sum over dimer coverings computable as the square root of a determinant — which makes the dimer model one of the few exactly solvable interacting lattice systems. In two dimensions, dimer models realize several conformal field theory (CFT) universality classes, classified by the central charge cc: the uniform square/rectangular-lattice dimer is described by a logarithmic CFT with c=−2c=-2 (the symplectic-fermion theory), while the anisotropic dimer model on the honeycomb lattice — solved by Wu (1968) and Nagle (1973), and often called the Kasteleyn model — has a c=1c=1 Gaussian (height-field) description. The latter exhibits a KDP-type transition ('potassium dihydrogen phosphate'-like, after the ferroelectric that realizes the same Landau theory): a transition driven by the proliferation of frozen, fully ordered configurations rather than by symmetry breaking, with a square-root specific-heat singularity at the phase boundaries.

The generalized checkerboard lattice is a square lattice whose four-site unit cell carries alternating horizontal weights x1,x2x_1,x_2 and vertical weights y1,y2y_1,y_2. Three inequivalent assignments of these weights to the checkerboard pattern are denoted checkerboard A, B, and C; checkerboard A was introduced by Kasteleyn, the thermodynamic-limit behavior of these and related geometries was studied by Wu, Lin, Sacco, Hsue, Yokoi, Salinas and others, and Cohn, Kenyon, and Propp treated the special case M=NM=N for checkerboard C. Izmailian, Hu, and Kenna (2015) derived the exact closed-form partition functions of all three models on the 2M×2N2M\times2N torus with periodic boundary conditions, as finite products over the four twisted sectors (α,β)(\alpha,\beta) (the 'spin structures' of the torus, (α,β)∈{(0,0),(0,12),(12,0),(12,12)}(\alpha,\beta)\in\{(0,0),(0,\frac{1}{2}),(\frac{1}{2},0),(\frac{1}{2},\frac{1}{2})\}), together with integral representations of the free energy. These formulas interpolate between limiting regimes: when x1=x2x_1=x_2 and y1=y2y_1=y_2, all three models reduce to the uniform rectangular-lattice dimer (c=−2c=-2 class); when one of the vertical weights y1y_1 or y2y_2 vanishes, checkerboard B reduces to the honeycomb 'K2' model (a vanishing horizontal weight x1x_1 or x2x_2 instead gives a one-dimensional strip), and when any one of the four weights vanishes, checkerboard C reduces to the honeycomb 'K1' model, both in the c=1c=1 Kasteleyn class with KDP-type transitions at y1=x1+x2y_1=x_1+x_2 (and permutations), where the specific heat has a square-root singularity (K1, and checkerboard C) and the second derivatives of the free energy have an inverse-root singularity (K2, and checkerboard B). The B and C lattices thus realize, within one exactly solvable family, a crossover between a c=−2c=-2 and a c=1c=1 regime.

The 2015 paper supplies the finite-lattice formulas but performs no asymptotic analysis of them; its introduction and abstract state that the results 'will be useful for further study on finite-size corrections and scaling' and 'provide insight into corrections to scaling arising from the finite-size versions of the model'. For the checkerboard-A lattice this extraction program was then completed by Izmailian, Wu, and Hu (2016) using the Kronecker-double-series technique of Ivashkevich, Izmailian, and Hu — the standard device, based on Euler–Maclaurin summation in the complex plane and asymptotics of Jacobi θ\theta functions, that converts finite-torus products into 1/S1/S expansions (with S=4MNS=4MN the number of sites). They obtained the exact asymptotic expansion of ln⁡Z\ln Z at the A-lattice critical point — the point t=0t=0, i.e. x1=x2x_1=x_2 and y1=y2y_1=y_2, where t2=[(x1−x2)2+(y1−y2)2]/(4x1x2)t^2=[(x_1-x_2)^2+(y_1-y_2)^2]/(4x_1x_2) is the natural 'temperature-like' variable, on the slice z=1z=1 (equal geometric means of horizontal and vertical weights, z2=y1y2/(x1x2)z^2=y_1y_2/(x_1x_2)) — to second order, exact finite-size corrections for the free energy, internal energy, specific heat, and higher derivatives, finite-size scaling functions (computed numerically from the exact partition function), the anomalous result that the shift exponent is λ=∞\lambda=\infty (the finite-size specific-heat pseudocritical point, i.e. the location of the finite-system maximum, coincides with the infinite-system critical point), the N→∞N\to\infty cylinder expansion, and the conclusion that both a c=−2c=-2 and a c=1c=1 conformal description are consistent with the finite-size data. The A lattice, however, has no honeycomb/KDP limit, so the crossover and KDP questions do not arise for it. For the checkerboard-B lattice, Chen, Hu, Izmailian, and Wu (2019) performed a numerical finite-size study based on the exact partition function: they analyzed the specific heat and the Fisher zeros (the zeros of the partition function in the complex plane of a weight or temperature-like variable, whose approach to the real axis signals the thermodynamic singularity), finding ring-, concentric-circle-, radial-segment-, and 'arabesque'-like zero patterns, a number of specific-heat peaks and zero circles that grows with system size, and strong anisotropy effects with a shift exponent λ\lambda generally unequal to 1/ν1/\nu (where ν\nu is the correlation-length exponent). That work explicitly confined itself to checkerboard B, leaving checkerboard C untreated, and provides no closed-form asymptotic expansions or analytic crossover scaling functions. What still does not exist in the literature is the analytic layer: the exact asymptotic finite-size expansions for B and C, crossover scaling functions between the c=−2c=-2 and c=1c=1 regimes, and an explicit analytic theory of the finite-size rounding and shift of the KDP-type singularities of these two lattices.

Problem Statement

From the exact closed-form products and free-energy integral representations of Izmailian, Hu, and Kenna (2015) for the dimer model on the 2M×2N2M\times2N checkerboard-B and checkerboard-C lattices with periodic boundary conditions, derive the analytic finite-size theory of these two models, comprising:

(i) the exact asymptotic expansion of ln⁡Z\ln Z (equivalently, the free energy) for large M,NM,N at fixed aspect ratio ρ=M/N\rho=M/N and general weights x1,x2,y1,y2x_1,x_2,y_1,y_2 — the analog, for checkerboards B and C, of the expansions obtained for checkerboard A in 2016 — giving the leading and subleading finite-size correction amplitudes, together with the induced finite-size corrections to thermodynamic quantities such as the internal energy and specific heat;

(ii) finite-size scaling and crossover scaling functions describing how finite tori interpolate between the c=−2c=-2 rectangular regime (x1=x2x_1=x_2, y1=y2y_1=y_2) and the c=1c=1 Kasteleyn (honeycomb K1/K2) regimes as the weights are varied; and

(iii) the explicit finite-size characterization of the KDP-type singularities: how the square-root specific-heat singularity (checkerboard C/K1) and the inverse-root singularities of the second derivatives of the free energy (checkerboard B/K2) at the phase boundaries y1=x1+x2y_1=x_1+x_2 (and permutations) are rounded and shifted at finite M,NM,N, including the approach of the Fisher zeros to the real axis.

These three components jointly serve one objective — converting the exact finite-lattice formulas into the complete analytic finite-size description of the two checkerboard geometries that exhibit the c=−2↔c=1c=-2\leftrightarrow c=1 crossover — and a complete answer must reduce, in the appropriate limits, to the rectangular-lattice behavior at x1=x2x_1=x_2, y1=y2y_1=y_2 (the c=−2c=-2 class) and to the known K1/K2 honeycomb thermodynamics, the reductions verified in the 2015 source.

References

  1. N. Sh. Izmailian, Chin-Kun Hu, and R. Kenna, 'Exact solution of the dimer model on the generalized finite checkerboard lattice', Phys. Rev. E 91, 062139 (2015). DOI: 10.1103/PhysRevE.91.062139, https://doi.org/10.1103/PhysRevE.91.062139
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